Is the data given in the two statements, labeled (1) and (2), sufficient for answering the question?All numbers used are real numbers.

1. Is −x < (1/x) ?(1) x < −1(2) |x| > 0

Ο Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.Ο Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.Ο BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Ο EACH statement ALONE is sufficient.Ο Statements (1) and (2) TOGETHER are NOT sufficient.AnswerStatement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.Statement (1) specifies that x < −1, which implies −x > 1, −1 < (1/x) < 0 and −x > (1/x).Thus, statement (1) ALONE is sufficient.Statement (2) specifies that |x| > 0, which impliesx > 0 or x < 0.If x > 0, then −x is negative, (1/x) is positive and −x < (1/x).If x < 0, then −x is positive, (1/x) is negative and −x > (1/x).Thus, statement (2) ALONE is not sufficient.

3.triangleWhat is the area of Δ ABC?(1) Δ ABC is isosceles.(2) AB= 10 and BC = 12

Ο Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.Ο Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.Ο BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Ο EACH statement ALONE is sufficient.Ο Statements (1) and (2) TOGETHER are NOT sufficient.AnswerBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Statement (1) only tells us that Δ ABC is isosceles which is not enough to find the area of the triangle. Thus, statement (1) ALONE is not sufficient.Statement (2) specifies the length of the base, but does not specify the altitude (or height) of the triangle. Thus, statement (2) ALONE is not sufficient.triangleIn an isosceles triangle, the altitude from the vertex angle bisects the base (as shown in the figure alongside). So, BD = ½ BC. Knowing ABand BD, the altitude AD can be found by the Pythagorean theorem. Since area of triangle = ½ base x altitude, the area can be determined as ½ BC x AD. Thus, BOTH statements TOGETHER are sufficient.

6.squareWhat is the perimeter of square KLMN?(1) KL = 4 inches(2) Its diagonal is √32 inches.

Ο Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.Ο Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.Ο BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Ο EACH statement ALONE is sufficient.Ο Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

EACH statement ALONE is sufficient.To find the perimeter of square, its side must be known becausePerimeter of square = 4 x Side.Statement (1) specifies the side. Thus, statement (1) ALONE is not sufficient.Statement (2) specifies the diagonal. The side of the square may be found using the Pythagorean theorem and then the perimeter calculated as follows:KL2 + LM2 = (√32)2Since KL = LM, 2LM2 = 32 or LM = √16 = 4 inches.Perimeter of square = 4 x 4 inches = 16 inches.Thus, statement (2) ALONE is sufficient.

8. A company prepares a consignment by packing 20 compact discs in each small box. If the same consignment was to be packed in big boxes with 50 discs in each box, how many big boxes would be required ?(1) The consignment is for a total of 6000 compact discs.(2) The ratio of the number of big boxes to the number of small boxes is 2 : 5.

Ο Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.Ο Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.Ο BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Ο EACH statement ALONE is sufficient.Ο Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.Let sand b be the number of small and big boxes, respectively.Then, total number of discs t = 20s = 50b.Statement (1) specifies t = 6000, which gives b = 6000/50.Thus, statement (1) ALONE is sufficient.Statement (2) specifies b/s = 2/5; however, this information is already known from 20s = 50b. Thus, statement (2) ALONE is not sufficient.

11.semicircleWhat is the length of side RS in the right-angled triangle PQR in the figure above ?(1) x = 30(2) Line segment PS has length 5.

Ο Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.Ο Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.Ο BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Ο EACH statement ALONE is sufficient.Ο Statements (1) and (2) TOGETHER are NOT sufficient.

Ο Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.Ο Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.Ο BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Ο EACH statement ALONE is sufficient.Ο Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

EACH statement ALONE is sufficient.Since the angles of a triangle add to 180o, ∠ A + ∠ B + ∠ C = 180o.Statement (1) states that ∠ B = ∠ C. Since∠ B = 36o, ∠ A may be calculated from∠ A = 180o − 2 ∠ B.Thus, statement (1) ALONE is sufficient.Statement (2) states that two sides of the triangle are equal in length,which implies the triangle is isosceles with ∠ B = ∠ C.As before, given ∠ B = 36o, ∠ A may be calculated from ∠ A = 180o − 2 ∠ B.Thus, statement (2) ALONE is sufficient.Therefore, EACH statement ALONE is sufficient.